Abstract: Consider the cyclic group C 2 of order two acting by complex-conjugation onthe unit circle S^1. The main result is that a finitely dominated manifold W ofdimension > 4 admits a cocompact, free, discontinuous action by the infinitedihedral group D \infty if and only if W is the infinite cyclic cover of a freeC 2-manifold M such that M admits a C 2-equivariant manifold approximatefibration to S^1. The novelty in this setting is the existence ofcodimension-one, invariant submanifolds of M and W. Along the way, we developan equivariant sucking principle for certain orthogonal actions of finitegroups on Euclidean space.